On differentiability of implicitly defined function in semi-parametric profile likelihood estimation

TL;DR

This paper investigates the differentiability of functions defined implicitly within semi-parametric profile likelihood estimation, offering an alternative approach to existing methodologies by utilizing direct expansion techniques.

Contribution

It demonstrates the applicability of a direct expansion method for analyzing differentiability in semi-parametric models, providing an alternative to existing parametrization approaches.

Findings

Establishes conditions for differentiability of implicitly defined functions

Shows the effectiveness of direct expansion in semi-parametric estimation

Provides theoretical foundation for alternative profile likelihood analysis

Abstract

In this paper, we study the differentiability of implicitly defined functions which we encounter in the profile likelihood estimation of parameters in semi-parametric models. Scott and Wild (Biometrika 84 (1997) 57-71; J. Statist. Plann. Inference 96 (2001) 3-27) and Murphy and van der Vaart (J. Amer. Statist. Assoc. 95 (2000) 449-485) developed methodologies that can avoid dealing with such implicitly defined functions by parametrizing parameters in the profile likelihood and using an approximate least favorable submodel in semi-parametric models. Our result shows applicability of an alternative approach presented in Hirose (Ann. Inst. Statist. Math. 63 (2011) 1247-1275) which uses the direct expansion of the profile likelihood.